Question 9.CS.1: Motor-Belt-Drive Shaft Design for Steady Loading A motor tra...

Reactions at bearings. From Equation 1.15, the torque applied by the pulley to the motor shaft equals

$kW =\frac{F_t V}{1000}=\frac{T n}{9549}$          (1.15)

$T_{A C}=\frac{9549 P}{n_0}=\frac{9549(55)}{4500}=116.7  N \cdot m$

The forces transmitted through the belt is therefore

$F_2-\frac{F_1}{5}=\frac{T_{A C}}{r}=\frac{116.7}{0.051}=2288  N$

or

$F_1=2860  N \quad \text { and } \quad F_2=572  N$

Applying the equilibrium equations to the free-body diagram of the shaft (Figure 9.4b), we have

The results indicate that $R_{A}$ and $R_{B}$ act in the directions shown in the figure.

Principal stresses. The largest moment takes place at support $B$ (Figure 9.4c) and has a value of

$M_B=3432(0.07)=240.2  N \cdot m$

Inasmuch as the torque is constant along the shaft, the critical sections are at $B$. It follows that

$\begin{array}{c} \tau=\frac{16 T}{\pi D^3}=\frac{16(116.7)}{\pi D^3}=\frac{1867.2}{\pi D^3} \\ \sigma_x=\frac{32 M}{\pi D^3}=\frac{32(240.2)}{\pi D^3}=\frac{7686.4}{\pi D^3} \end{array}$

and $\sigma _{y}$ = 0. For the case under consideration, Equation 3.33 reduce to

$\sigma_{\max , \min }=\sigma_{1.2}=\frac{\sigma_x+\sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{x y}^2}$          (3.33)

$\begin{aligned} \sigma_{1,2} &=\frac{\sigma_x}{2} \pm \sqrt{\left(\frac{\sigma_x}{2}\right)^2+\tau^2} \\ &=\frac{3843.2}{\pi D^3} \pm \frac{1}{\pi D^3} \sqrt{\frac{(7686.4)^2}{4}+(1867.2)^2} \\ &=\frac{1}{\pi D^3}(3843.2 \pm 4272.8) \end{aligned}$

from which

$\sigma_1=\frac{8116}{\pi D^3} \quad \sigma_2=-\frac{429.6}{\pi D^3}$     (b)

Energy of distortion theory of failure. Through the use of Equation 6.14,

$\left[\sigma_1^2-\sigma_1 \sigma_2+\sigma_2^2\right]^{1 / 2}=\frac{S_y}{n}$

This, after introducing Equation (b), leads to

$\frac{1}{\pi D^3}\left[(8116)^2-(8116)(-429.6)+(-429.6)^2\right]^{1 / 2}=\frac{390\left(10^6\right)}{3.5}$

Solving,

$D$ = 0.0288 m = 28.8 mm

Comment: A commercially available shaft diameter of 30 mm should be selected.